## Abstract

This work presents an on-design component-level multiple-objective optimization of a small-scaled uncooled cavity-stabilized combustor. Optimization is performed at the maximum power condition of the engine thermodynamic cycle. The computational fluid dynamics simulations are managed by a supervised machine learning algorithm to divide a continuous and deterministic design space into nondominated Pareto frontier and dominated design points. Steady, compressible three-dimensional simulations are performed using a multiphase realizable k–ε RANS and nonadiabatic flamelet/progress variable combustion model. Conjugate heat transfer through the combustor liner is also considered. There are fifteen geometrical input parameters and four objective functions viz., maximization of combustion efficiency, and minimization of total pressure losses, pattern factor, and critical liner area factor. The baseline combustor design is based on engineering guidelines developed over the past two decades. The small-scale baseline design performs remarkably well. Direct optimization calculations are performed on this baseline design. In terms of Pareto optimality, the baseline design remains in the Pareto frontier throughout the optimization. However, the optimization calculations show improvement from an initial design point population to later iteration design points. The optimization calculations report other nondominated designs in the Pareto frontier. The Euclidean distance from design points to the Utopic point is used to select a “best” and “worst” design point for future fabrication and experimentation. The methodology to perform computational fluid dynamics optimization calculations of a small-scale uncooled combustor is expected to be useful for guiding the design and development of future gas turbine combustors.

## Introduction

Small-scale gas turbine combustors are important for many applications. Burning all of the fuel in a small combustor introduces considerable challenges. The fuel must atomize, vaporize, mix, and burn with the air in a shorter residence time when compared to mid- or large-scale gas turbine engines. In addition, the combustor must meet engine design criteria such as combustion efficiency, pressure loss, profile and pattern factor, and liner temperature requirements. It is experimentally and computationally unfeasible to perform a full factorial design study because of the plethora of geometric parameters that can be adjusted to optimize combustor performance.

Conventional gas turbine combustors typically have been designed through *ad hoc* and heuristic guidelines based on trial-and-error experimentation. This iterative combustor design process raises the development cost. Original engine manufacturers (OEMs) utilize computational fluid dynamics (CFD) extensively to guide some of their combustor design processes. Nevertheless, CFD has been limited to a few sample simulations that may not be representative of the entire design space. In turn, this leads to a local optimal design that may not represent the global optimum.

Autonomous software can be coupled within a framework to create design points that are inputs to a cad software that dynamically creates combustor geometries. The geometry can then be passed to the meshing software to independently create meshes, which are sent to the solver to perform CFD simulations. The results are postprocessed by another software to compute output parameters of interest. The output parameters are automatically passed to a direct optimization software that makes a decision on whether or not to add more design points.

We have previously shown [1–6] that a combustor can be improved from its baseline using machine learning techniques. Our experience has suggested that it is more appropriate to optimize the combustor in either the engine or the test section in which the combustor will be tested. In this study, we optimize the combustor at maximum engine power condition in a virtual test section that represents the actual test section. Because this approach does not consider engine component-to-component interactions, we refer to it as component-level design optimization. In contrast, if the combustor were to be optimized in the engine, which is beyond the scope of this research, it would be considered a system-level design optimization. Design optimization in a test section offers several advantages. First, the combustor with accurate hole discharge coefficient is optimized, and design point geometries are ready for fabrication. Second, the boundary-induced disturbances that may affect the combustor flow field are mitigated. Third, because the combustor is in the test section, the simulations provide more realistic flow splits; one single mass flow rate is specified, and the solution determines the flow splits throughout the combustor.

Furthermore, our previous investigation on design optimization [6] suggested that varying the cooling hole sizes and count on the liner to target certain cooling flow rate is not feasible or the outcomes outweigh the benefits for the small-scale combustor. Because the residence time is short in the small-scale combustor, removing the effusion cooling holes, which are difficult for implementing optimization, is reasonable if we include conjugate heat transfer and minimize the critical liner area factor that could adversely affect liner durability.

The main purpose of this investigation is to optimize a small-scale outboard cavity combustor with four objective functions (i.e., maximization of combustion efficiency, and minimization of pattern factor, critical liner area factor, and total pressure losses). The specific objectives of the optimization are: (1) to develop a decision support process to select a nondominated design point from the Pareto front (“best” design) and a dominated design point (“worst” design) for future higher fidelity CFD simulations and experimental testing and (2) to compare the features of the “best” and “worst” design points with the baseline design in terms of velocity streamlines, temperature, equivalence ratio, liner temperature, and global combustor performance.

## Physical Model

This section describes details related to the combustor parametric geometry, computational domain, governing equations, combustion model, surrogate model, and objective functions.

### Combustor Parametric Geometry.

The geometry of the outboard cavity combustor is illustrated in Fig. 1. The combustor contains several air jet groups to induce recirculation zones in the cavity, enhance fuel/air mixing, increase flame stability, and dilute high temperature zones. There are 48 cavity forward driver jets, 48 cavity aft driver jets, 8 outer liner jets, and two rows of 8 staggered inner liner jets. Because of the periodicity of the combustor, only a 45 deg sector is modeled. The inlet conditions for all air jets are prescribed to total (stagnation) pressure of 396 kPa, total temperature of 497 K, turbulence intensity of 5%, and ratio of turbulent-to-laminar viscosity of 10. The outlet (fuel and air) mass flow rate ($m\u02d9exit$) is 0.653 kg/s for the full annular combustor. The CFD solver automatically computes mass flow rate distributions through the various air liner holes. The global equivalence ratio (*ϕ*) is 0.344.

The fifteen geometric input parameters are illustrated in Fig. 1 and described in Table 1. These input parameters are chosen based on a combination of engineering judgment and previous experimental studies of trapped vortex combustors [1–11]. There are thirteen air inlet parameters including jet diameters, jet locations, chute aspect ratio and inclination angle; one parameter for the cavity axial length; and one parameter for fuel injector location. In contrast with our previous work [1–6], there is only one inlet mass flow rate and one pressure outlet boundary condition.

Input parameters | Baseline (DP0) | Lower bound | Upper bound | |
---|---|---|---|---|

P1 | Cavity forward driver jet radial location | 6.24 | 4.2+P4/2 | 6.5−P4/2 |

P2 | Cavity aft driver jet radial location | 4.74 | 4.4+P5/2 | 6.5−P5/2 |

P3 | Cavity length | 3.31 | 1.7 | 5.0 |

P4 | Cavity forward driver jet diameter | 0.37 | 0.2 | 2·π·P1/48 − 0.2 |

P5 | Cavity aft driver jet diameter | 0.43 | 0.2 | 2·π·P2/48 − 0.2 |

P6 | Fuel injection radial location | 6.01 | 4.2 | 6.4 |

P7 | Inner dilution jet diameter | 0.89 | 0.2 | 1.3 |

P8 | Outer dilution jet diameter | 0.89 | 0.2 | 1.3 |

P9 | Inner/Outer dilution jet axial location | 3.21 | 0.6+(P7+P8)/4 | 7.0−P3−P8/2 |

P10 | Chute angular width | 30.2 | 4.2 | 31.9 |

P11 | Chute inclination angle | 20.0 | −20 | 20 |

P12 | Chute height | 0.71 | 0.2 | 1.3 |

P13 | Chute radial location | 3.64 | 2.8+P12/2 | 4.1−P12/2 |

P14 | Forward dilution jet axial location | 5.98 | 7.1−P3+P15/2 | 8.3−P15/2 |

P15 | Forward dilution jet diameter | 0.84 | 0.20 | 1.3 |

Input parameters | Baseline (DP0) | Lower bound | Upper bound | |
---|---|---|---|---|

P1 | Cavity forward driver jet radial location | 6.24 | 4.2+P4/2 | 6.5−P4/2 |

P2 | Cavity aft driver jet radial location | 4.74 | 4.4+P5/2 | 6.5−P5/2 |

P3 | Cavity length | 3.31 | 1.7 | 5.0 |

P4 | Cavity forward driver jet diameter | 0.37 | 0.2 | 2·π·P1/48 − 0.2 |

P5 | Cavity aft driver jet diameter | 0.43 | 0.2 | 2·π·P2/48 − 0.2 |

P6 | Fuel injection radial location | 6.01 | 4.2 | 6.4 |

P7 | Inner dilution jet diameter | 0.89 | 0.2 | 1.3 |

P8 | Outer dilution jet diameter | 0.89 | 0.2 | 1.3 |

P9 | Inner/Outer dilution jet axial location | 3.21 | 0.6+(P7+P8)/4 | 7.0−P3−P8/2 |

P10 | Chute angular width | 30.2 | 4.2 | 31.9 |

P11 | Chute inclination angle | 20.0 | −20 | 20 |

P12 | Chute height | 0.71 | 0.2 | 1.3 |

P13 | Chute radial location | 3.64 | 2.8+P12/2 | 4.1−P12/2 |

P14 | Forward dilution jet axial location | 5.98 | 7.1−P3+P15/2 | 8.3−P15/2 |

P15 | Forward dilution jet diameter | 0.84 | 0.20 | 1.3 |

The radial locations of forward (P1) and aft (P2) driver jets, fuel injection (P6), and chutes (P13) are listed with respect to the axis of rotation. The axial location of the dilution jets (P9 and P14) is relative to the combustor exit plane. Locations and diameters are in cm and angles are in deg.

The input parameters' upper and lower bounds are spread as far as possible, constrained by manufacturing limitations. Some input parameter bounds exhibit constant values, whereas other bounds depend on another input parameter. The bounds ensure creation of realizable geometries and maximize the design space. For example, the positioning of the outer liner dilution jets (P9) must not collide with the cavity axial length (P3). Hence, when the total cavity length is larger, the outer liner dilution jet range is more restricted. On the other hand, when the cavity axial length (P3) is shorter, the outer liner dilution jet (P9) has more space to move axially.

### Computational Domains and Meshes.

The computational domain is an assembly of five individual computational zones as illustrated in Fig. 2. They are the upstream and downstream plenums, the (engine shroud) enclosure, the combustor, and the liner. The former four computational zones are fluids whereas the latter represents a solid zone. The upstream and downstream plenum meshes are static throughout the optimization procedure. The enclosure, combustor, and hardware are meshed for each design point. The hardware liner thickness is fixed at 0.5 mm. All computational zones are meshed independently from each other, utilizing cut-cell meshing at the boundaries. Therefore, the five meshes are aggregated in the solver and are nonconformal with each other. The periodicity of the five zones is the same, and five zones are all nonconformal. Adaptive mesh refinement with four levels of refinement is performed on an initial grid with maximum cell size of 0.64 mm [3]. These settings yield grid independent results [3].

### Governing Equations.

Compressible steady multiphase three-dimensional simulations of a small-scale outboard cavity combustor are performed using the realizable *k*–*ε* RANS model with a nonadiabatic diffusion flamelet/progress variable combustion model. Turbulence-chemistry interaction is tabulated a priori in a tri-dimensional table as a function of mixture fraction, mixture fraction variance, and progress variable. Beta- and double-delta-presumed probability density functions (PDFs) are used for the mixture fraction and progress variable, respectively. The products of these marginal PDFs are employed for the joint probability of low-dimensional manifold variables. This is used to convolute the thermochemical properties and any Favre-averaged thermochemical variable such as temperature and species mass fractions. A single-component surrogate species (n-C_{12}H_{26}) is used as fuel [12], which in turn is injected as a hollow liquid spray. The discrete phase model allows for the liquid to exchange mass and momentum, while latent heat of vaporization is considered negligible. Second-order accurate schemes are used for the spatial discretization.

### Direct Optimization Model.

Adaptive multiple-objective (AMO) optimization is used to find the Pareto frontier of the design space and to identify optimal designs. ansysworkbench [13], designxplorer [14], designmodeler [15], ansysmeshing [16], fluent [17,18], and cfd-post [19] are utilized.

The AMO approach combines a Kriging response surface (RS) with the multiple objective genetic algorithm. The Kriging RS reduces the number of required CFD simulations by taking advantage of the cheap RS evaluations. The initial sample size is set to 48, which leads to sample independent results [4]. These are generated by a Latin hypercube sampling (LHS) algorithm. In multiple objective genetic algorithm, the crossover combines chromosomes (i.e., sequence of input parameters) of two parents to obtain two offspring. The reasoning behind crossover is that the recombination of input parameters of two parent design points can lead to a superior offspring design point. After the offspring design, points have been generated they could undergo mutation. This alters one or more input parameters (i.e., chromosomes) from their initial state. An offspring that displays a new input parameter value within its chromosomes ultimately enhances the gene pool. The mutation may also lead to a better solution by avoiding stagnation at a local extremum.

Adaptive multiple-objective checks for the error of each new design point with the current RS. If the error is acceptable, the design point is evaluated on the RS. If it is not acceptable, the process (i.e., geometry and mesh generation, CFD simulation, and objective function computation) is repeated to improve the RS. Subsequently, a convergence criteria is checked. Convergence is achieved when either a percentage of current design points fall within the Pareto frontier or an output parameter stability criteria is met (in terms of mean and standard deviation). The metasimulation also stops if a maximum number of iterations is reached. If the metasimulation does not converge before this, it is considered nonconvergent. Nondominated sorted genetic algorithm-II [20] is used for sorting the nondominated Pareto frontier. More detailed information of AMO can be found elsewhere [1–3,6].

### Objective Functions.

Four objective functions are presented in Table 2. The average exit temperature ($T4,avg$) is maximized in order to increase combustion efficiency, $\eta $. We also want to minimize the maximum exit temperature ($T4,max$) because as it increases, the pattern factor (PF) also increases, which adversely affects the downstream turbine stator and rotor. Further considerations for this constraint can be found in Ref. [3]. The critical liner area factor ($Acritical$) must be minimized in order to diminish metal yielding occurrence and increase hardware durability. The total pressure losses ($TPL$) are minimized because it is always a goal to achieve high mixing and high combustion efficiency with a low pattern factor at low cost (i.e., low total pressure loss). It is the designer desire that the optimum designs outperform the baseline design (DP0) in terms of these four objective functions.

Indices | Output parameter | Objective type |
---|---|---|

1 | Average exit temperature ($T4,avg$) | Maximize |

2 | Maximum exit temperature ($T4,max$) | Minimize |

3 | Critical liner area factor (A_{critical}) | Minimize |

4 | Total pressure loss ($TPL$) | Minimize |

Indices | Output parameter | Objective type |
---|---|---|

1 | Average exit temperature ($T4,avg$) | Maximize |

2 | Maximum exit temperature ($T4,max$) | Minimize |

3 | Critical liner area factor (A_{critical}) | Minimize |

4 | Total pressure loss ($TPL$) | Minimize |

### Decision Support Process.

The ideal total pressure losses for the reacting flow design space is approximated as $TPLideal,space\u2009\u2248min(TPL)reacting,space$.

The station numbering is based on the Society of Automotive Engineers Aerospace Recommended Practice 755 [21].

## Results and Discussion

The optimization results and the comparison between design points are presented in the subsequent sections.

### Baseline Computational Fluid Dynamics Results.

The reacting flow field in terms of streamlines, equivalence ratio, and temperature of the baseline design point (DP0) is presented in Figs. 3–6. The combustor cavity exhibits a large circumferential piecewise counterclockwise recirculation region. This recirculation region is only observed at the 0 deg and 11.25 deg planes. The cavity recirculation region is no longer evident at the 22.5 deg plane. The streamlines indicate that there are several recirculation regions in the region between the engine casing and combustor liner (i.e., enclosure zone). There are two recirculation regions upstream of the forward combustor wall that restricts radially inward flow toward the chutes and the inner liner. There is a recirculation region immediately downstream the cavity aft wall. There are two larger low velocity recirculation regions downstream of the main inner and outer dilution jets. Contrary to our previous work [5], the forward driver jet is tilted radially inward when it enters the combustor. This suggests that the forward driver jet discharge coefficient is lower when the engine casing is considered than when it is not considered. This, in turn, redistributes the mass flow rate through the other liner holes because the total air mass flow rate through the combustor is fixed. Similarly, the chute exhibits a larger outer recirculation region that reduces its discharge coefficient and alters the flow distribution through the various liner holes. The result also suggests that combustors are sensitive to the enclosure in which they are designed. Therefore, combustor optimization should at least consider the engine casing to better account for the flow distribution and discharge coefficients.

The fuel is injected axially at the centerplane and at a radial location slightly inward from the forward driver jet location. The chute prevents rapid mixing of the fuel with the air as indicated by the saturated red color in Fig. 4. However, the fuel mixes with air in the circumferential direction as indicated by the “greenish” and “yellowish” colors of the 11.25 deg and 22.5 deg planes of Fig. 4. According to these results, intense fuel burning occurs anywhere in between fuel injections and downstream the chute.

There is clearly a direct correlation between equivalence ratio and temperature as illustrated in Fig. 5. High temperatures are found in between the fuel injection sites (i.e., 22.5 deg plane) as well as downstream the chutes (cf., 0 deg plane). In addition, there is heat transfer from the combustor to the enclosure zone. Heat is transferred to the inner downstream regions of the enclosure zone as well as immediately downstream the cavity wall. The latter appears to slightly preheat the aft driver jet. Also the flame attaches to the chute outer wall due to the presence of a recirculation region. This recirculation region does not appear in our earlier work which does not include the engine casing [5].

The combustor liner temperature is presented in Fig. 6. The temperature depicted in this figure is that of the inner side wall surface of the solid hardware zone. Overall the temperature of the liner does exceed the 1300 K threshold. A few hot regions are observed on the cavity aft wall as well as immediately upstream and downstream of the main inner liner dilution hole.

The qualitative and quantitative performance of the baseline combustor is acceptable for a small-scale uncooled combustor. It was designed following previous knowledge of experimental, numerical, and analytical studies of cavity combustors [9]. The exit average temperature is 1293 K corresponding to a combustion efficiency of 93.4%. The maximum exit temperature is 1388 K corresponding to a pattern factor of 0.12. The critical liner area factor is 3.1%. The total pressure losses from station 3.1 to station 4 are 6.9%. Contrary to our previous work [1–6], the total pressure losses are greater (than *ca.* 4%) for two reasons. First, we include the engine casing enclosure. Second, the enclosure zone exhibits substantial recirculation regions that increase total pressure losses.

### Optimization Results.

Direct optimization calculations are performed. The optimization simulation converged after 11 iterations and 64 CFD calculations because all design points in the last iteration are (within 2%) in the Pareto frontier according to the AMO algorithm. There are nine RS evaluations corresponding to the second and tenth iteration. The baseline design point does not explicitly participate in the optimization procedure. The optimization does not provide a design point that demotes the baseline design point to a dominated point.

Spearman's order-rank correlation is performed for the input and output parameters. A 95% confidence level with a student's *t*-test is also evaluated for these correlations. The combustion efficiency is negatively correlated with forward driver jet radial location and positively correlated with the inner/outer dilution jet axial location. The pattern factor is positively correlated with the inner dilution jet diameter. The critical liner area factor is positively correlated with the fuel injector radial location. The total pressure losses are positively correlated with the chute inclination angle. According to the order-rank correlation, a better combustor performance can be achieved by decreasing the forward driver jet radial location and moving the inner/outer dilution jets toward the cavity, while reducing the inner dilution jet diameter and fuel injector radial location and chute inclination angle.

Table 3 illustrates the dominated^{2} and nondominated design point average and standard deviation input and output parameters. An open source code [22] was utilized to sort out the Pareto frontier utilizing nondominated sorting genetic algorithm-II. It is expected that the standard deviation of the nondominated design points be smaller than that of the dominated design points. This would suggest that the optimization is converging to a set of design configurations. The standard deviation for nondominated design points decreases from that of the dominated design points except for the fuel injection radial location, inner/outer dilution jet axial location, and critical liner area factor. The forward cavity driver jet radial location standard deviation decreases from 10% (dominated) to 8% (nondominated). The average and standard deviation of aft driver jet radial location remain nearly constant. The average cavity length is converging at nearly 3.2 cm as indicated by the reduction in standard deviation from 27% to 24%. In order to mitigate the effect of the enclosure lower discharge coefficients of the forward driver jets, the nondominated design points place these jets radially outward in comparison with the dominated design points (5.9 cm versus 5.6 cm). According to the global Spearman's order-rank correlation, this could adversely affect the combustion efficiency. Nondominated design points also improve the air flow rate through the aft drive jets by increasing the hole diameter (0.4 cm versus 0.3 cm). Evidently, nondominated designs prefer inner and outer dilution jet diameters with equal sizes, while dominated design points exhibit larger inner dilution jets than outer dilution jets. There is, nonetheless, uncertainty in the location of these jets as suggested by a standard deviation of 59%. According to the global Spearman's order-rank correlation placing the inner/outer dilution jets near the cavity improves combustion efficiency. The nondominated design points prefer larger height-to-width chutes than dominated design points. The chute radial location and inclination are about the same for both dominated and nondominated design points. However, both dominated and nondominated design points exhibit large uncertainty or standard deviations for the chute inclination angle. Spearman's order rank correlation indicates that tilting the chutes radially inward reduces the total pressure losses. The nondominated design points prefer forward dilution jets with larger diameter and closer to the cavity as indicated by their average values and reduced standard deviations. The nondominated design points exhibit about 10% improvement in combustion efficiency, more than 100% improvement in pattern factor and critical liner area factor, and reduced total pressure losses.

There are many contradictions between the nondominated design point features and the Spearman's order-rank correlation. This could be attributed to an under-resolved design space or to the fact that some features vary more in percentage than other features. Hence, Spearman's order-rank correlation could be attributing large output parameter sensitivities to those input parameters that very more (in percentage) throughout the current design space.

The results in terms of output parameters of the objective functions and Euclidean distance ($d(dp,up)$) to the Utopic design point are illustrated in Fig. 7. There are two blowout cases from these simulations (i.e., $dp=8$ and $dp=25$). Figure 7 indicates that the combustion efficiency, excluding the two blowout cases, is as low as *ca.* 60%. AMO shows no much difference in combustion efficiency between the initial population ($dp\u226448$) and the last iteration ($dp>48$). However, the pattern factor clearly indicates substantial improvement between the last iteration and the initial population. Considerable improvement is also observed for the critical liner area factor, which reaches values as high as *ca.* 40% for the initial population and then drops to less than 20% for the last iteration. Significant reduction in pressure losses is also observed between the initial population and last iteration in terms of total pressure losses. For the former, the total pressure losses are as large as 20% and then drop to *ca.* 7% in the last iteration. It is evident that AMO is improving the overall combustor design from the initial population.

Figure 7 also shows that the range of Euclidean distance decreases from the initial population to the last iteration. This confirms that overall combustor design improvements are made through the optimization process. The Pareto frontier is indicated with magenta dots. The Pareto frontier nondominated design points also exhibit shorter Euclidean distance to the Utopic point. DP0 exhibits the shortest Euclidean distance to the Utopic point followed closely by DP58, DP38, and DP43 (in ascending distance). These four design points are the most promising small-scale uncooled combustors for future higher fidelity CFD simulations, experimental testing, or both.

### Dominated “Worst” Design Point Computational Fluid Dynamics Results.

Generally, this step is not required. However, we select a “worst” design point from the dominated design points because we plan to experimental test our findings in future work. A dominated design point with comparable cavity length to the baseline is selected to maintain a similar combustor volume. Another criteria for the “worst” design point is to select a design with radially outward fuel jet injection as the baseline. In this sense, the effect of hole diameters and locations can be compared with the other designs at relatively fixed volume and fuel residence time. In addition, this dominated design point should exhibit a Euclidean distance somewhere in between the blowout and Pareto frontier Euclidean distance (*cf.* Fig. 7). The reasoning behind this decision is to be able to select a “worst” design point that could sustain a flame. For these reasons, DP40 is chosen with a cavity length of 3.15 cm (versus 3.33 cm for the baseline) and fuel radial location of 5.3 cm (versus 6 cm for the baseline).

The reacting flow field in terms of streamlines, equivalence ratio, and temperature of the “worst” design point is presented in Figs. 8– 10. It is evident that the flow field in the enclosure zone is qualitatively similar to that of the baseline with the same number of large recirculation regions. However, the difference is that because the aft driver jets are placed radially outward the recirculation region downstream the aft cavity wall is now nonexistent. It should be expected less heat transfer from the combustor cavity to the enclosure through this wall. Again the forward driver jet is tilted radially inward as with the baseline. The driver jet configuration leads to two larger recirculation regions in the cavity. In contrast to the baseline design point, these two recirculation regions represent uninterrupted toroidal vortices. The chute also promotes two larger recirculation regions adjacent to the forward combustor wall that only persist at the chute circumferential location.

The axially injected fuel does not penetrate into the cavity as with the baseline design point (*cf.* Fig. 9). Here, the chute does not “seal” the cavity effectively, and fuel is transported radially inward along the forward cavity wall. Evidently, the fuel does not mix well as indicated by the red contours displayed outside the combustor cavity. According to these results, most of the burning would occur downstream of the chutes.

There is again a direct correlation between equivalence ratio and temperature as illustrated in Fig. 10. High temperatures are found mostly at the centerplane downstream the chutes. In addition, there is heat transfer from the combustor to the enclosure zone. Heat is transferred primarily through the combustor outer liner downstream of the aft cavity wall. This suggests that the main outer dilution jet is preheated. The combustor temperature flow field is characterized by very high temperature and low temperature regions. In contrast, the baseline exhibits more gradual temperature gradients (*cf.* Fig. 5). This again indicates that the fuel and air are not well mixed for DP40.

The combustor liner temperature is presented in Fig. 11. The temperature depicted in this figure is that of inner side wall surface of the solid hardware cell zone. The combustor liner temperature exhibits critical areas at the cavity outer wall and both downstream inner and outer liner. The latter, however, exhibits temperatures well above the threshold temperature. Therefore, the durability of the downstream outer liner is probably shortened with respect to that of the baseline.

The qualitative and quantitative performance of the “worst” combustor is poor. The exit average temperature is 1242 K equivalent to combustion efficiency of 87.3%. The maximum exit temperature is 1860 K equivalent to a pattern factor of 0.72. The critical combustor liner area is 18%. The total pressure losses from station 3.1 to station 4 are 9.6%. The baseline design point (DP0) clearly performs better than the “worst” design point in terms of the four objective functions (*cf.* Table 2).

### Nondominated “Best” Design Points Computational Fluid Dynamics Results.

The reacting flow fields in terms of streamlines, equivalence ratio, and temperature of the three “best” design points (DP58, DP38, and DP43) are presented in Figs. 12– 15. DP58, DP38, and DP43 are within the Pareto frontier and exhibit the lowest Euclidean distance (in ascending order) to the Utopic design point (*cf.* Fig. 7). These three designs exhibit comparable cavity axial lengths (3.77, 3.22, and 3.52 cm for DP58, DP38, and DP43, respectively). The fuel injector radial locations are 6.1, 6.2, and 5.5 cm, respectively. For DP58, the fuel injector is aligned with the forward driver jets. For DP38, the fuel injector is slightly radially outward with respect to the forward driver jets. For DP43, the fuel injector is radially inward with respect to the forward driver jet. The flow field in the enclosure zone is relatively similar for all combustors. There is at least one large recirculation region downstream of the cavity aft wall. There is a flow constriction caused by two recirculation regions upstream the forward cavity wall. There is a smaller recirculation region adjacent to the downstream inner liner. Inside the combustor cavity, DP58 and DP38 exhibit two radially outward counter-rotating recirculation regions that are circumferentially unimpeded. On the other hand, DP38 exhibits a single counterclockwise recirculation zone along the azimuthal direction. For the three design points, the main dilution jets impinge against each other, and there is a high velocity region at the center between the inner and outer liner. The dilution jets for all these designs are closer to the combustor exit plane (station 4) than they are to the axial location of the cavity aft wall. The hole diameters are similar for the inner and outer dilution jets. The forward inner liner dilution jet momentum penetrates into the cavity flow field for DP43. This may disrupt the double recirculation structure, turning it into a single recirculation region. The forward inner liner dilution jets do not penetrate into the cavity for DP58 and DP38.

The equivalence ratio distributions in Fig. 13 reveal similarities between the “best” design points. The chute closes off the cavity such that the fuel injected into the cavity is “sealed” from the mainstream flow. This forces the fuel to mix circumferentially, which in turn increases the residence time and leads to better mixing. The cavities for these three designs exhibit fuel-rich equivalence ratio regions inside the cavity at the centerplane, where the fuel is circumferentially injected. Fuel then mixes with air circumferentially. This observation is consistent with the baseline design results (cf. Fig. 4). In contrast, the “worst” design is characterized by fuel-rich equivalence ratio regions in the mainstream flow because the chute does not “seal” the cavity effectively (cf. Fig. 9). Then, the fuel is transferred from the cavity radially inward with diminished circumferential convection and diffusion. Hence, intense burning occurs at circumferential locations aligned with the fuel injection azimuthal angle (cf. Fig. 10). Better performance is achieved when burning occurs in a flow field with smaller temperature gradients (cf. Figs. 5, 10, and 14). For this to occur, the fuel that is injected with infinite equivalence ratio (outside the flammable regime) has to mix with air and when the fuel reaches the fuel-rich flammability limit. The “worst” design is burning near stoichiometric at large regions in the flow field reaching temperatures of 2400 K. The “best” designs and the baseline reach temperatures of near 2100 K, which suggests off-stoichiometric burning. The latter designs are also characterized for higher temperature regions in between fuel injector sites, whereas the former design is characterized by higher temperature regions at the fuel injector azimuthal angle.

The combustor liner temperature is presented in Fig. 15. Overall the wall temperature distributions of these designs are good. Nonetheless, DP58 exhibit less critical liner area factor concerns than DP38 and DP43. Even though DP38 is closer to the Utopic point than DP43 is, the former exhibits greater regions of hardware durability concerns than the former. This indicates that solely the Euclidean distance as discriminator between better and worse designs within the Pareto frontier is sometimes not sufficient and the application helps to make a decision.

The qualitative and quantitative performance of the “best” design points is acceptable for small-scale uncooled combustors. For DP58, the exit average temperature is 1279 K equivalent to a combustion efficiency of 91.7%. The maximum exit temperature is 1409 K, equivalent to a pattern factor of 0.17. The critical liner area factor is 1.4%. The total pressure losses from station 3.1 to station 4 are 6.3%. For DP38, the exit average temperature is 1322 K, equivalent to combustion efficiency of 96.7%. The maximum exit temperature is 1428 K, equivalent to a pattern factor of 0.13. The critical liner area factor is 12.9%. The total pressure losses from station 3.1 to station 4 are 11%. For DP43, the exit average temperatures are 1325 K equivalent to combustion efficiency of 97.1%. The maximum exit temperature is 1426 K, equivalent to a pattern factor of 0.12. The critical liner area factor is 13.8%. The total pressure losses from station 3.1 to station 4 are 13.3%. Therefore, DP38 and DP43 outperform DP58 in terms of combustion efficiency and pattern factor, but the latter outperforms the former designs in terms of liner critical area and total pressure losses. DP38 and DP43 are results of the LHS design of experiments for the initial population, whereas the latter is a result of the converged AMO optimization algorithm.

### Design Point Selection Summary.

After detailed analyses and considerations, it is proposed that three combustor designs (DP0, DP40, and DP58) be further studied with higher fidelity CFD and experimental testing. These three design points represent the baseline, the “worst,” and the “best” design points, respectively. The “best” design point exhibits the smallest Euclidean distance (*cf.* Fig. 7) within the converged surrogate-model algorithm. In addition, it is representative of the nondominated design point Pareto frontier (*cf.* Table 3). Moreover, the qualitative results indicate a combustor flow field consistent with other nondominated design points and the baseline design point. The “worst” design point exhibits a larger Euclidean distance (*cf.* Fig. 7) within the initial LHS population. This design is expected to ignite and stabilize a flame. In addition, it is representative of the dominated design points (*cf.* Table 3). Furthermore, the qualitative results indicate a different combustor flow field for the “best” and baseline design points. Table 4 summarizes the input and output parameters for these three design points.

Parameter | Baseline (DP0) | “Worst” (DP40) | “Best” (DP58) |
---|---|---|---|

P1 | 6.24 | 6.1 | 6.2 |

P2 | 4.74 | 6.2 | 6.3 |

P3 | 3.31 | 3.2 | 3.8 |

P4 | 0.37 | 0.5 | 0.5 |

P5 | 0.43 | 0.3 | 0.4 |

P6 | 6.01 | 5.3 | 6.2 |

P7 | 0.89 | 1.3 | 0.5 |

P8 | 0.89 | 0.3 | 0.7 |

P9 | 3.21 | 1.7 | 1.2 |

P10 | 30.2 | 17.9 | 12.5 |

P11 | 20.0 | −1.3 | −1.8 |

P12 | 0.71 | 0.3 | 0.7 |

P13 | 3.64 | 3.1 | 3.5 |

P14 | 5.98 | 7.1 | 7.6 |

P15 | 0.84 | 0.3 | 1.3 |

$T4,avg$ | 1293 | 1242 | 1279 |

$T4,max$ | 1388 | 1860 | 1409 |

$\eta $ | 93.4 | 87.3 | 91.7 |

$PF$ | 0.12 | 0.72 | 0.17 |

$Acritical$ | 3.1 | 18 | 1.4 |

$TPL$ | 6.9 | 9.6 | 6.3 |

Parameter | Baseline (DP0) | “Worst” (DP40) | “Best” (DP58) |
---|---|---|---|

P1 | 6.24 | 6.1 | 6.2 |

P2 | 4.74 | 6.2 | 6.3 |

P3 | 3.31 | 3.2 | 3.8 |

P4 | 0.37 | 0.5 | 0.5 |

P5 | 0.43 | 0.3 | 0.4 |

P6 | 6.01 | 5.3 | 6.2 |

P7 | 0.89 | 1.3 | 0.5 |

P8 | 0.89 | 0.3 | 0.7 |

P9 | 3.21 | 1.7 | 1.2 |

P10 | 30.2 | 17.9 | 12.5 |

P11 | 20.0 | −1.3 | −1.8 |

P12 | 0.71 | 0.3 | 0.7 |

P13 | 3.64 | 3.1 | 3.5 |

P14 | 5.98 | 7.1 | 7.6 |

P15 | 0.84 | 0.3 | 1.3 |

$T4,avg$ | 1293 | 1242 | 1279 |

$T4,max$ | 1388 | 1860 | 1409 |

$\eta $ | 93.4 | 87.3 | 91.7 |

$PF$ | 0.12 | 0.72 | 0.17 |

$Acritical$ | 3.1 | 18 | 1.4 |

$TPL$ | 6.9 | 9.6 | 6.3 |

The input parameter definitions are described in Table 1. Units are in cm, degrees, Kelvin, and %.

## Conclusions

This work demonstrates methods for performing component-level multi-objective optimization of a small-scale uncooled outboard cavity combustor. The work is motivated by the need for numerical methods to guide the design, development, and optimization of modern small-scale gas turbine combustor.

The baseline combustor design is founded on engineering procedures developed over the past two decades. The small-scale baseline design performs remarkably well in terms of combustion efficiency, pattern factor, critical liner area factor, and total pressure losses. In terms of Pareto optimality, the baseline design remains in the Pareto frontier throughout the optimization calculations.

Optimization is performed at the on-design engine maximum power operation. The optimization calculations conclusively show improvement from an initial design point population to later iteration design points.

The Euclidean distance from design points to the Utopic point is used as a decision support process to select a “best” and “worst” design point for future higher fidelity CFD calculations and/or experimental testing. The methodology to perform CFD optimization calculations of a small-scale uncooled combustor is expected to be useful for guiding the design and development of future gas turbine combustors.

## Acknowledgment

We thank Markus Rumpfkeil from UDRI, and Joshua Sykes and Timothy Gallagher from ISSI for discussing the results. Special thanks to Scott Stouffer from UDRI for providing the resources and Nathan Thomas from UDRI for postprocessing the results.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.

## Funding Data

U.S. Air Force Research Laboratory (Agreement No. FA8650-15-D-2505; Funder ID: 10.13039/100006602).

## Nomenclature

## Footnotes

They are called dominated because all output parameters of interested (objective function) are outperformed by another combustor design. Conversely, nondominated term refers to the fact that at least one objective function is not outperformed by any other combustor design.